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296                        ANALYTICAL MECHANICS
3.   A gun can project a shot to a height of ~, where R is the radius of the earth.   Taking the variation of the gravitational force with altitude, show that the gun can command -5 of the earth's surface.
4.   A particle is projected into a smooth horizontal circular groove. The particle is attracted towards a point in the radius which joins the
position of projection with the center, with a force equal to ^  Show that in order that the particle may be able to make complete revolutions the initial velocity must not be less than~~~77> where a is the radius of the
groove and b the distance of the center of force from the center of the groove.
5.   A comet describing a parabolic orbit about the sun collides with a body of equal mass at rest.   Show that the center of mass of the two describes a circle about the sun as center.
6.   Prove that the least velocity with which a bod}'' must be projected from the north pole so as to hit the surface of the earth at the equator is about 4-J miles per second, and that the angle of elevation is 22.5.
7.   A particle moves in the common Held of two fixed centers of force of equal intensity.   The particle is attracted towards one of the centers with a force which varies as its distance from that center, and repelled from the other center according to the same law.   Show that the orbit is a parabola.
8.   A particle moves in a field in which the force is repulsive and varies inversely as the square of the distance from the center of force.   Show that the orbit is a hyperbola.
9.   In the preceding problem show that the radius vector sweeps over equal areas in equal intervals of time.